282 CHEMICAL ENGINEERING
Theory of Settlement in Circular Tanks and Their Capacity. — By circular settling tanks are meant ones which are fed at the center and overflow around the whole periphery. Where there is absolute free settling, a condition which will be approximated if the proportion of solids is so small as to have no appreciable effect on the density of the stream of material entering a circular tank, then the path of a particle entering the center of the tank will be given by the equation
_ y ~
Further conditions for satisfying this equation are that the particles entering the tank will fall from the point of entry to the bottom with a finite definite rate. Second that there is uniform flow in the tank from the center to the periphery and parenthetically a uniform rate of flow of all liquid particles from top to bottom towards the periphery. Under this conception all particles in vertical lines at any given distance from the center will after any given flow interval still be in vertical lines and have arrived at the same distance from the tank center. Perfect flow of this kind is impossible to obtain. The entering stream can be forced to the bottom of the tank but tends to rise to the surface and flow in all directions towards the periphery. If it is endeavored to distribute the entering stream uniformly from top to bottom at its entry point some success can be had by employing suitably arranged deflectors but after leaving the deflectors the current will tend to rise towards the overflow. The position of the overflow governs the direction of outflow of the material fed to a tank. In a deep circular tank with overflow at the periphery at the top, the bottom areas particularly at the periphery are stagnant. The last assumption under the formula is that all the entering stream passes out into the body through the vertical center line, that it is concentrated at this line before outward flow and that the particles whose path is sought may flow outwardly along this vertical line. At whatever point they leave it will be the origin of the curve. Where comparison of the settling capacity of various shaped tanks is to be made it is convenient to assume that the particle begins its outward path at the surface of the water and that the origin is at this point or better that this is the origin of outward movement of all the particles entering the various shaped tanks.
In the formula m is the downward rate of subsidence which may be taken for water from the accompanying logarithmic curve for galena and quartz. Merely read off the number corresponding to the log in a logarithmic table and the velocity V and diameter of grain D will be given in millimeters per second and millimeters. Other substances of different specific gravity will have full velocities proportional to the specific gravity of the two given, nearly (Cf. " Velocity of Galena and Quartz Falling in Water," Trans. A. I. M. E., XXXVIII). R is the radius of the tank and V the velocity at the overflow. To obtain V the cubical contents Q of the entering stream in cubic feet per second is divided by the depth of water in the tank times 2-n-R. The depth of the water can be obtained by adding to the depth of the tank at the overflow
point the head of water obtained from the expression h = \/Q )L which is merely
\ O^iTTrC ff
the theoretical weir formula for discharge with the quantity Q transposed and the member 62, the width of the slot squared, converted into 2-n-R2.
The expression for the path of a particle in a rectangular tank is very simple:
y equals — . The head of water at the discharge end can be computed from the theoretical formula for discharge.